Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

This paper is a continuation of our previous works about coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It concerns a study on the properties of wavefunctions in the phase space representation and the momentum dispersion operator, its representations and eigenvalue equation. After the recall of some results from our previous papers, we give most of the main properties of the phase space wavefunctions and consider some examples of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any phase space wavefunction is solution of this equation.


Introduction
The present work can be considered as part of a series of studies related to a phase space representation of quantum theory introduced and developed in [1], [2] and [3]. Because of the uncertainty relation [4], the problem of considering phase space, which mix momentum with coordinate, in quantum theory is an interesting challenge. Many works related to this subject have been already performed. We may quote for instance [5][6][7][8][9][10][11][12]. Many of these works are based on the approach introduced by Wigner in [5].
In the reference [1], we have considered an approach using the current formulation of quantum mechanics based on linear operators theory and Hilbert space. Our previous work [1] may be considered as extension in the framework of quantum theory of the results obtained in [14]. The phase space representation that we have defined is based on the introduction of quantum states, denoted | , , , 〉. These states are defined by the means values , and statistical dispersions (variance) ∆ and ∆ of coordinate and momentum being a positive integer). We have the relations : is a Hermite polynomial of order . It has been established that a state | , , , 〉 is an eigenstate of the momentum dispersion operator Σ < = ℶ 2 and the coordinate dispersion operator Σ > = ℬ ℶ 2 . The explicit expressions of these operators are and the corresponding eigenvalues equations are E @ | , , , 〉 = 2 + 1 ℬ| , , , 〉 @ | , , , 〉 = 2 + 1 | , , , 〉 As the eigenvalues of the momentum and coordinate dispersion operators are proportional and as they have the same eigenstates, it is sufficient to consider only the momentum dispersion operator ℶ 2 .
In the reference [2], we have tackled the problem of finding the representations of coordinate, momentum and dispersion operators in the frameworks of the phase space representation. We have established that they can be at the same time represented both with matrix and differential operators in the basis {| , , , 〉} defining this phase space representation. In this paper, our goal is to perform a study on the properties of wavefunctions in the phase space representation and their relation with the eigenvalue equation of the momentum dispersion operator ℶ 2 . We show and verify explicitly, in particular, that these wavefunctions are as expected the eigenfuctions of the differential operator which represents ℶ 2 .

Phase Space Representation of Momentum Dispersion Operator
Let us consider the momentum dispersion operator ℶ 2 . It can be put in the form in which and on the other hand the differential operators representations From the relation (8) and (9), we obtain for the matrix representation of the operator ℶ 2 ℶ 2 J = , , , |ℶ 2 |K, , , and from the relation (8) and (11), we obtain for the differential operator representation Using the relation (7), (12), (13) and (1), we obtain respectively for the matrix and differential operator representations of the momentum dispersion operator ℶ 2 ℶ J 2 = , , , |ℶ 2 |K, , , We may remark that the expression of ℶ 2 J corresponds to the fact that the elements of the basis {| , , , } defining the phase space representation are the eigenvectors of ℶ 2 .

Phase Space Wavefunctions
A phase space wavefunction Ψ , , of a particle is a phase space representation of its quantum vector state |T . It is equal to the inner product of the vector |T with an element of the basis {| , , , } defining the phase space representation. As established in our papers [1], In which, the functions T = |T and T X = |T are the wavefunctions corresponding to the state |T respectively in coordinate and momentum representations. The functions " * , , , and " # * , , , are the complex conjugates of the wavefunctions" , , , and " # , , , corresponding to the state | , , , respectively in coordinate and momentum representations. The expressions of " and " # are given in (3) and (4). There are two possibilities to expand a state |T in the basis {| , , , } of the phase space representation. The first one corresponds to any fixed value of , and and the expansion is obtained by varying the positive integer : The second one corresponds to any fixed value of and and the expansion is obtained by varying the real number and In the framework of the probabilistic interpretation of quantum mechanics, these relations permit us to give the following physical interpretation of the phase space wavefunctions: 1. The relations (17) and (21) allow to interpret the functions |Ψ , , | as the probability to find a particle in a state | , , , , for a fixed value of , and knowing that the state of this particle is |T .
2. The relations (18) and (22)  as the density of probability to find a particle in a state | , , , , for a fixed value of and , knowing that the state of this particle is |T .
It follows in particular that, for |T = | ′, ′, ′, ′ , we have the relation Let us now consider any state |T . It results from (28) that in the basis {| , , , } we have

Differential Equation Satisfied by Phase Space Wavefunctions
On one hand, from the relations (14) and (17), it can be deduced that for any phase space wavefunctions Ψ , , = , , , |T we have the relation , , , |ℶ 2 |T =^ 2 + 1 ℬM J J Ψ J , , and on the other hand, from the definition of differential operator representation, as given in our work [2], we have in which ℶ Q 2 is the differential operator representation of the momentum dispersion operator ℶ 2 given in the relation (15).
It follows from the relations (33), (34) and (35) that any phase space wavefunction Ψ , , satisfies the differential equation This equation is also the eigenvalue equation for the differential operator representation ℶ Q 2 of the momentum dispersion operator ℶ 2 . And then, according to it, any phase space wavefunctions Ψ is an eigenfunction of ℶ Q 2 with the eigenvalue equal to 2 + 1 ℬ.
The equation (36) can be also checked explicitly using the properties (32) of the phase space wavefunctions. In fact, according to this relation we have This relation means that the function " * , , , is an eigenfunction of ℶ Q 2 with the eigenvalue 2 + 1 ℬ. Using the relation (38), we can deduce from (37) that Use of the relations (32), (35) and (39) permits to have, as expected, an explicit checking of (36).

Conclusion
In the phase space representation, the momentum dispersion operator ℶ 2 can be at the same time represented either with the diagonal matrix ℶ J 2 given in the relation (14) or with the differential operator ℶ Q 2 given in the relation (15). with eigenvalue equal to 2 + 1 ℬ. This result can be considered as being logically expected since the elements of the basis { | , , , } defining the phase space representation themselves are the eigenstates of ℶ 2 .
In the section 3, we have done some recall concerning the phase space wavefunctions and give most of their main properties. Among these properties, we may notice a particular one which is given in the relation (32). This relation shows that the values of a phase space wavefunction Ψ , , corresponding to different values of the parameters and variables , , and can be linked using the function b c , , ; `, `, ` defined in the relation (23). As discussed in the last part of the section 4, this property of phase space wavefunction given in the relation (32)  Using this function, we obtain the relation